Lesson 1 of 3 6–9% of the exam

You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

By Daniil Tarakanov Updated Oct 16, 20262026 curriculum

Expected value, variance and standard deviation of a forecast

Weight each possible outcome by its probability and add: that is the expected value. Variance weights each squared distance from the expected value by its probability, and the standard deviation is its square root. These are forecasts of the future, so there is no n − 1.

6 min cram, 26 min full read. Based on the 2026 CFA® Program Level I curriculum, Quantitative Methods, Learning Module 4. This is the 2026 version of the lesson. Sitting in February 2027 or later? Read the updated 2027 version.

Tailor this to you: take the self-assessment

Before this: Statistical Measures of Asset Returns

On the exam

Calculate expected values, variances, and standard deviations and demonstrate their application to investment problems

Calculate expected values, variances and standard deviations from a probability distribution, and apply them to an investment decision. Expect a three-scenario table where the simple average and the n − 1 formula are answer options.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

What is an expected value?

An expected value of a random variable is the probability-weighted average of its possible outcomes. Take each outcome, multiply by its probability, and add. It looks forward: it is a forecast built from a probability distribution , a list of outcomes with the probability of each, summing to 1. Laying out the outcomes as a short table of scenarios, such as recession, normal and boom, is scenario analysis .

Expected value Memorize

E(X) = P(x_1)\,x_1 + P(x_2)\,x_2 + \cdots + P(x_n)\,x_n = \sum_{i=1}^{n} P(x_i)\,x_i

Rearrange itExpected value: solve for E(X) or for a missing outcome

The question asks for

A forecast gives −12% with probability 0.25, 8% with probability 0.5 and 20% with probability 0.25. The expected value is closest to:

Probability of scenario 1Outcome in scenario 1%Probability of scenario 2Outcome in scenario 2%Probability of scenario 3Outcome in scenario 3%

E (X) = P_{1} x_{1} + P_{2} x_{2} + P_{3} x_{3}

0.25 × (−12) + 0.5 × 8 + 0.25 × 20 = 6.00%

Expected value, E(X)6.00%

On your calculator

0.25×12+|-+(0.5×8)+(0.25×20)=

+|- change sign

0.25ENTER12CHS×0.5ENTER8×+0.25ENTER20×+

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

The 6% is a balance point: if the outcomes were weights on a seesaw, the board would balance at 6. It is not the return you should expect in any one year. No scenario returns exactly 6%.

Exam tip On the BA II Plus in chain mode, wrap each product in brackets after the first one. Otherwise 0.25 × 12 + 0.5 is evaluated before the 0.5 is multiplied.

How do you measure how uncertain a forecast is?

The variance of a random variable is the probability-weighted average of the squared deviations from the expected value. The standard deviation is its square root, in the same units as the outcomes, so it is the figure to quote.

Variance of a forecast Memorize

\sigma^2(X) = \sum_{i=1}^{n} P(x_i)\,[x_i - E(X)]^2

Standard deviation of a forecast Memorize

\sigma(X) = \sqrt{\sigma^2(X)}

Worked example The risk of the same share

Continue with E = 6%.

Deviations. −12 − 6 = −18, 8 − 6 = +2 and 20 − 6 = +14.
Square and weight. 0.25 × 324 = 81, 0.50 × 4 = 2 and 0.25 × 196 = 49.
Variance. 81 + 2 + 49 = 132 (%²).
Standard deviation. √132 = 11.49%.

Standard deviation on your calculator

2NDDATA2NDCLR WORK open and clear the worksheet
12+|-ENTER↓25ENTER↓ X01 = −12 with weight 25, the probability typed as a percent
8ENTER↓50ENTER↓
20ENTER↓25ENTER
2NDSTAT2NDSET until 1-V shows
↓↓ X̄ = 6, the expected value
↓↓ σx = 11.4891, the probability-weighted figure (not Sx)
x² 132, the variance
2NDQUIT

2ND second function (the shifted label above a key) · DATA data worksheet · CLR WORK clear the worksheet · +|- change sign · ENTER enter (on the HP, puts the number on the stack) · STAT statistics worksheet · SET change the setting · x² square · QUIT leave the worksheet

fCLEAR Σ
12CHSENTER0.25Σ+
8ENTER0.5Σ+
20ENTER0.25Σ+
gx̄w 6, the expected value
18ENTER×0.25× 81
2ENTER×0.5×+ 83
14ENTER×0.25×+ 132, the variance
g√x 11.49

f gold shift key · CLEAR Σ clear the statistics registers · CHS change sign · ENTER enter (on the HP, puts the number on the stack) · Σ+ add a data point · g blue shift key · x̄w weighted mean · √x square root

Rearrange itVariance and standard deviation of a forecast: solve for either

The question asks for

A forecast gives −12% (probability 0.25), 8% (0.5) and 20% (0.25), with an expected value of 6.00%. The variance, in %², is closest to:

Probability of scenario 1Outcome in scenario 1%Probability of scenario 2Outcome in scenario 2%Probability of scenario 3Outcome in scenario 3%Expected value, E(X)%

σ^{2} = i \sum P_{i} [x_{i} - E (X)]^{2}

0.25 × (−12 − 6)² + 0.5 × (8 − 6)² + 0.25 × (20 − 6)² = 132 (%²)

Variance (in %²)132

On your calculator

(12+|-−6)x²×0.25+((8−6)x²×0.5)+((20−6)x²×0.25)=

+|- change sign · x² square

12CHSENTER6−ENTER×0.25×8ENTER6−ENTER×0.5×+20ENTER6−ENTER×0.25×+

CHS change sign · ENTER enter (on the HP, puts the number on the stack)

InteractiveExpected value: where the scenarios balance

Try this 0 of 3

Show the equal-weighting trap and compare it with E(R)
Change only the boom outcome until the expected return is 8%
Keep E(R) at 6% but bring the standard deviation below 8%

Each block sits at its outcome; its height is its probability. The triangle is the balance point, E(X). The shaded band runs from one standard deviation below it to one above.

Recession: probability0.25

Return%

Normal year: probability0.50

Return%

Boom: probability0.25

Return%

Show the equal-weighting trap (simple average)

Expected return, E(R)6.00%

Variance, σ²132.00 (%²)

Standard deviation, σ11.49%

The beam balances at 6.00%. Recession adds the most to the variance (61% of it); a typical miss from the balance point is about 11.49 percentage points.

Show working

E(R) = 0.25 × (−12) + 0.5 × 8 + 0.25 × 20 = (−3) + 4 + 5 = 6.00%.

Deviations from E(R): −18, +2, +14.

σ² = 0.25 × (−18)² + 0.5 × 2² + 0.25 × 14² = 81 + 2 + 49 = 132.00 (%²).

σ = √132.00 = 11.49%.

These are forecasts, so the probabilities are the weights: no n − 1 anywhere.

Expected value and σ on your calculator

2NDDATA2NDCLR WORKclear old data
12+|-ENTER↓25ENTER↓X01, Y01 = probability × 100
8ENTER↓50ENTER↓X02, Y02 = probability × 100
20ENTER↓25ENTER↓X03, Y03 = probability × 100
2NDSTATpress 2ND SET until 1-V shows
↓n = 100, the sum of the weights
↓X̄ = 6.00, the expected value
↓Sx: ignore it, it divides by n − 1
↓σx = 11.49, the standard deviation
x²132.00, the variance

fCLEAR Σ
12CHSENTER0.25Σ+outcome, then its probability
8ENTER0.5Σ+outcome, then its probability
20ENTER0.25Σ+outcome, then its probability
gx̄w6.00, the expected value
18CHSENTER×0.25×2ENTER×0.5×+14ENTER×0.25×+132.00, the variance
g√x11.49, the standard deviation

On the TI the probabilities go in as whole-number frequencies (× 100); read σx, not Sx. The HP 12C has no weighted standard deviation, so the variance is summed term by term.

The expected value is a balance point, weighted by how likely each outcome is.

Drag a probability up and watch the balance point move toward its outcome. Spread the outcomes apart and the shaded band widens even though the expected value does not change.

Why is there no n − 1 here?

In the previous module the data were past returns, so the mean was estimated from the sample and the deviations were averaged with n − 1. Here nobody has estimated anything from a sample. The probabilities are your forecast of how often each outcome occurs, and they already do the averaging.

	Historical data (LM3)	Forecast with probabilities (this lesson)
Center	Sample mean, equal weights	Expected value, weights = probabilities
Variance	Σ(X − X̄)² ÷ (n − 1)	Σ P × (X − E)²
Standard deviation	Square root of the variance	Square root of the variance

Notice the shape is the same: a weighted average of squared deviations. Equal weights of 1/n (or 1/(n − 1)) are replaced by the probabilities.

How do you use these in a decision?

Compare the expected value for reward and the standard deviation for risk. Suppose stock B returns 4% or 8%, each with probability 0.5. Its expected return is 6%, the same as the share above, but its standard deviation is only 2%. Two forecasts with the same reward can carry very different risk, and a risk-averse investor prefers the lower standard deviation. The same arithmetic applies to any forecast: earnings per share, revenue growth, project cash flows or credit losses.

Go deeper Why a fair bet can still be a bad idea

A coin flip that wins 100 or loses 100 has an expected value of zero, so it is “fair”. It still has a standard deviation of 100. A risk-averse investor needs a positive expected return to accept that spread, which is why an investment’s reward has to be judged against its risk and not on its own. Portfolio mathematics in the next module builds on exactly these two numbers.

Go deeper Why this variance is the one used in portfolios

The next module combines the variances and the covariance of several assets into a portfolio variance. Each of those inputs is a probability-weighted quantity of this form when it comes from a forecast, so the formula in this lesson is the building block.

Memorize this

MemorizeForecast formulas

Measure	Formula	Weights
Expected value	Σ P(x) × x	probabilities
Variance	Σ P(x) × (x − E)²	probabilities
Std. deviation	√variance	same units as x

Memory hook Same shapes as the historical formulas in LM3, with the weight 1/n (or 1/(n − 1)) swapped for the probability P.

Know it cold Forecast with probabilities: weight by P, never divide by n − 1. History: sample formulas with n − 1.

From a table of scenarios to risk and reward

Probability distribution of outcomeseach outcome with its probability; probabilities add to 1

Expected value Memorize Σ P × x, the probability-weighted average
- Reads as the forecast, not a promised outcome
Variance Memorize Σ P × (x − E)², units are squared
- Weights probabilities, so no n − 1
Standard deviation Memorize √variance, same units as the outcomes
- Reads as the typical distance from the forecast

Expected value

E(X) = P(x_1)\,x_1 + P(x_2)\,x_2 + \cdots + P(x_n)\,x_n = \sum_{i=1}^{n} P(x_i)\,x_i

Quick example

A return is 10% with probability 0.4 and 4% with probability 0.6. E = 0.4 × 10 + 0.6 × 4 = 6.4%.

On your calculator

0.4×10+(0.6×4)= 6.4

0.4ENTER10×0.6ENTER4×+ 6.4

ENTER enter (on the HP, puts the number on the stack)

Variance of a forecast

\sigma^2(X) = \sum_{i=1}^{n} P(x_i)\,[x_i - E(X)]^2

Quick example

Same forecast, E = 6.4. 0.4 × (10 − 6.4)² + 0.6 × (4 − 6.4)² = 5.184 + 3.456 = 8.64 (%²).

On your calculator

(10−6.4)x²×0.4+((4−6.4)x²×0.6)= 8.64

x² square

10ENTER6.4−ENTER×0.4×4ENTER6.4−ENTER×0.6×+ 8.64

ENTER enter (on the HP, puts the number on the stack)

Standard deviation of a forecast

\sigma(X) = \sqrt{\sigma^2(X)}

Quick example

Variance 8.64 (%²). √8.64 = 2.94%.

On your calculator

8.64√x 2.94

√x square root

8.64g√x 2.94

g blue shift key · √x square root

How it is asked

Expected value with unequal probabilities

Returns 15% (0.2), 8% (0.5), −5% (0.3). Expected return?

3.0 + 4.0 − 1.5 = 5.5%. Trap: 6.0% (simple average).
Standard deviation of a two-outcome forecast

10% and 2%, each with probability 0.5. Standard deviation?

E = 6, variance = 0.5 × 16 + 0.5 × 16 = 16, σ = 4.00%. Traps: 16 (variance), 5.66% (÷ n − 1).
Which formula applies?

Variance computed from forecast probabilities: how are the squared deviations treated?

Weighted by their probabilities. Not divided by n or n − 1.

Exam traps

Equal-weighting the outcomes. A simple average ignores that some scenarios are likelier than others.
Dividing the weighted squared deviations by n − 1. The probabilities already do the averaging.
Reporting the variance when the question asks for the standard deviation (the variance is in %²).
Treating the expected value as the most likely outcome. It can be a value that never occurs.

Checkpoint

4 exam-style questions. About 90 seconds each.

1An analyst forecasts earnings per share of INR 2.40 (probability 0.3), INR 2.00 (probability 0.5) or INR 1.50 (probability 0.2). Expected EPS is closest to:
2For the same forecast (INR 2.40 with probability 0.3, INR 2.00 with 0.5, INR 1.50 with 0.2), the standard deviation of EPS is closest to:
3When variance is computed from forecast probabilities rather than from a historical sample, the squared deviations from the expected value are:
4A fund manager assigns these probabilities to next year's return: −10% (0.1), 0% (0.4), 10% (0.3) and 20% (0.2). The standard deviation of the return is closest to:

Key takeaways

E(X) = Σ P × x: the probability-weighted average of the outcomes.
σ² = Σ P × (x − E)², and σ = √σ².
Probabilities are the weights, so a forecast has no n − 1.
Two forecasts with the same expected value can carry very different risk: compare σ.

Expected value

E(X) = P(x_1)\,x_1 + P(x_2)\,x_2 + \cdots + P(x_n)\,x_n = \sum_{i=1}^{n} P(x_i)\,x_i

Quick example

A return is 10% with probability 0.4 and 4% with probability 0.6. E = 0.4 × 10 + 0.6 × 4 = 6.4%.

On your calculator

0.4×10+(0.6×4)= 6.4

0.4ENTER10×0.6ENTER4×+ 6.4

ENTER enter (on the HP, puts the number on the stack)

Variance of a forecast

\sigma^2(X) = \sum_{i=1}^{n} P(x_i)\,[x_i - E(X)]^2

Quick example

Same forecast, E = 6.4. 0.4 × (10 − 6.4)² + 0.6 × (4 − 6.4)² = 5.184 + 3.456 = 8.64 (%²).

On your calculator

(10−6.4)x²×0.4+((4−6.4)x²×0.6)= 8.64

x² square

10ENTER6.4−ENTER×0.4×4ENTER6.4−ENTER×0.6×+ 8.64

ENTER enter (on the HP, puts the number on the stack)

Standard deviation of a forecast

\sigma(X) = \sqrt{\sigma^2(X)}

Quick example

Variance 8.64 (%²). √8.64 = 2.94%.

On your calculator

8.64√x 2.94

√x square root

8.64g√x 2.94

g blue shift key · √x square root

Next lesson Probability trees and conditional expected values

Questions about this lesson?

Where this comes back

Portfolio Mathematics Quantitative Methods

Questions candidates ask

How do you calculate expected return with probabilities?

Multiply each possible return by its probability and add the products: E(R) = ΣP × R. A 25% chance of −12%, a 50% chance of 8% and a 25% chance of 20% give −3 + 4 + 5 = 6%. The probabilities must add to 1.

What is the difference between sample variance and the variance of a probability distribution?

Sample variance averages squared deviations of past observations, equal-weighted, using n − 1. The variance of a probability distribution weights each squared deviation from the expected value by its probability. Forecasts use the second form, so there is no n − 1.

What does expected value mean in finance?

It is the probability-weighted average of the possible outcomes: a forecast of the long-run average, not a prediction of what will happen. A 6% expected return can come from a share that returns −12%, 8% or 20% and never exactly 6%.

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